VERTICALLY DIFFERENTIATED MONOPOLY WITH

A POSITIONAL GOOD

LUCA LAMBERTINI and RAIMONDELLO ORSINI*University of Bologna

We analyse positional effects in a monopoly market with vertical differentiation, comparing

monopoly and social planning. The provision of quality under monopoly depends upon the relative

size of positional effects and the hedonic evaluation of quality. An elitarian equilibrium where quality

increases in the level of positional concern emerges under monopoly, only if the market is sufficiently

rich. Under social planning, quality increases in the level of positional externality, independently of

market affluency. As long as partial market coverage obtains under both regimes, the monopoly

deadweight loss decreases as the positional externality becomes more relevant.

I . Introduction

The purchase of some goods can be due to social motivation rather than intrinsic

characteristics. Consumption behaviour can be classified according to the distinction between

material and positional goods (Hirsch, 1976; Frank, 1985a, b), or between functional and non-

functional motivation (Leibenstein, 1950).

In the quest for social distinction, individuals try to consume goods which, for their quality

and price or for their limited supply, cannot be purchased by all. In the existing literature on

positional or snob goods, the external effect in consumption due to social concerns is

generally modelled through a relative-consumption mechanism. The utility derived from

consumption is a function of the quantity purchased relative to the average of the society or

the reference group to whom the consumer compares. This approach tends to ignore that very

often the social distinction accrues not from the quantity purchased, but from the quality of the

chosen good. Positional externalities are analysed by Grilo et al. (2001) in a duopoly model of

spatial competition with full market coverage, which also allows to deal with vertical

differentiation.1

Here, in line with casual observation, we propose an alternative approach where the

consumption choice is dichotomous: some consumers buy the positional good, some others

do not and the externality precisely relies on the fact that there are consumers who cannot

afford to buy the positional good.2

* Correspondence: Raimondello Orsini, Dipartimento di Scienze Economiche, University of Bologna,Strada Maggiore 45, 40125 Bologna, Italy. Email: orsini@spbo.unibo.it1 The linear model of horizontal differentiation with convex transportation costs is a special case of avertical differentiation model with quadratic costs of quality improvements (Cremer and Thisse, 1991).Moreover, as in Grilo et al. (2001), if firms are allowed to locate also outside the linear segment alongwhich consumers are distributed, then there exist location pairs giving rise to vertical differentiation(Gabszewicz and Thisse, 1986; Tabuchi and Thisse, 1995; Lambertini, 1997a).2 The behaviour of a monopolist in a market where buyers’ satisfaction increases in the number ofconsumers excluded from purchase is analysed in Basu (1987), who suggests that the positional concernmay be traced back to either quality signalling by the firm or status seeking by consumers.

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In a partial equilibrium framework, we analyse the behaviour of a monopolist selling a

positional good. The utility derived from consumption by each individual is inversely related

to the number of other consumers buying the good. If consumers are sensitive to the size of

demand, the producer’s choice of the output level can be compared to the provision of product

quality, in that restricting output is essentially like providing consumers with a good

characterised by a superior quality, and conversely. Since output restriction is positively

evaluated by buyers, one might think that there is no reason to regulate a monopoly with

positional effects. However, it is also true that any output restriction involves an increase in

the number of consumers not being served, and, a priori, the net welfare effect of positional

concern is ambiguous. Therefore, the existence of status effects in markets where intrinsic

quality is endogenous prompts for a reassessment of the welfare distortion associated with

market power, as compared to the social optimum.

In order to provide a theoretical framework apt to evaluate this issues, we analyse the

monopoly case, building upon Spence’s (1975) seminal paper on vertical differentiation,

introducing a positional effect into consumer preferences. We investigate in detail a model

with standard assumptions concerning consumer distribution and technology. A single-

product monopolist supplies a good whose production entails a variable cost, which is

assumed to be convex in the quality level. Consumer distribution is uniform. In this setting, it

is known that, if positional effects were absent, a profit-maximising monopolist would supply

the same quality as a social planner, as long as partial market coverage obtains (Spence,

1975).3 We describe the behaviour of equilibrium prices, qualities and output levels, and the

resulting welfare implications, finding that the monopolist’s choices depend on the relative

size of marginal willingness to pay for intrinsic quality and positional effects. If buyers are

sufficiently rich, then an elitarian equilibrium emerges, where the number of buyers shrinks as

the positional concern becomes more relevant. On the other hand, if consumers’ valuation for

quality is below a critical threshold, the monopolist exploits the positional attribute of the

good, supplying a low quality to an increasing number of consumers as the positional

externality grows larger. In this case, the incentive to expand output increases in the extent of

the externality. In both cases, the welfare loss due to monopoly power shrinks as the role of

positional externalities in determining consumers’ satisfaction becomes more relevant. Profits

are always increasing in the amount of positional externality, while social welfare exhibits a

non-monotone behaviour.

The paper is organised as follows. Section II presents the model. The monopoly

equilibrium and the social optimum are investigated in Sections III and IV, respectively. They

are then comparatively evaluated in Section V. Section VI contains concluding remarks.

I I . Setup

Consider a monopoly market for a good whose utility depends on both intrinsic

characteristics, which are represented by quality q, and the social status which its purchase

confers to consumers. Social distinction arises because the market is covered only partially,

i.e. there are some agents who do not consume. Therefore, preferences exhibit a positional

externality, whose amount depends inversely on market demand x. Consumers are

characterised by parameter h, representing the individual marginal willingness to pay for

3 See also Sheshinski (1976), Mussa and Rosen (1978), Itoh (1983) and Champsaur and Rochet (1989).

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quality. Consumers are uniformly distributed over the interval ½�hh � 1; �hh�, with �hh � 1: The

number of individuals is normalised to 1. Accordingly, modelling social distinction is

economically meaningful only under partial market coverage (pmc; i.e., x < 1). The case of

full market coverage (fmc; i.e., x ¼ 1) collapses into the standard model without positional

externality (see Spence, 1975; Lambertini, 1997b).

Each consumer buys one unit of the good maximising the net surplus he obtains, provided

that it is non-negative

U ¼ hqþ að1 � xÞ � p ð1Þ

where p is the price charged by the monopolist, while a (the same for all the agents) is a

positive coefficient representing the weight of the positional externality in the utility function,

and x is the market demand for the good, so that 1 � x is the number of consumers who are

not served. This implies that, ceteris paribus, an additional purchase by a consumer

previously priced out of the market produces a negative externality on the group of

individuals who were already being served.

Demand is x ¼ �hh � hh, where hh is the marginal willingness to pay for quality of the marginal

consumer, i.e., the individual who is indifferent between buying or not. For this consumer, the

following equality must hold

UðhhÞ ¼ hhqþ að1 � �hh þ hhÞ � p ¼ 0; ð2Þ

yielding hhða; p; qÞ ¼ ðp þ a�hh � aÞ=ðqþ aÞ. Hence, under partial market coverage, market

demand is

x �hh � hhða; p; qÞ ¼�hhq� p þ a

qþ afor all p; q; af g

such that hhða; p; qÞ 2 ð�hh � 1; �hh� ðpmcÞ; ð3Þ

When max �hh � 1; hhða; p; qÞn o

¼ �hh � 1; full market coverage ( fmc) obtains, i.e., x ¼ 1; and

the positional externality disappears. This accounts for the fact that, when all consumers buy,

the purchase of the good does not yield any social distinction. The shape of the demand

function is illustrated in Figure 1.

From (3) and Figure 1, we notice that the direct market demand becomes flatter as the

positional concern increases. That is, the absolute value of the price elasticity of demand is

decreasing in a. However, a priori, there is no reason to expect a monotone relationship

between the welfare loss due to monopoly power and the price elasticity of demand (or a).4

On the supply side, production involves total costs C ¼ q2x: There are constant returns to

scale and unit production costs cðqÞ are convex in quality. The profit function is then

PM ¼ p � q2� �

x; ð4Þ

where superscript M stands for monopoly. Except for the presence of positional effects, this

setup replicates the model introduced by Spence (1975). He proves that, for a given output

4 The price elasticity of demand is, in absolute value

ep�� �� ¼ � @x

@p px¼ p

�hhq� p þ a

which is decreasing in a. On the lack of a univocal relationship between the welfare loss generated bymonopoly power and demand elasticity, see Tirole (1988, p. 67).

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level, if consumers are uniformly distributed and the market is covered only partially, the

monopolist supplies the socially optimal quality. In the following we will show that this is not

the case when a positional concern is present.

I I I . Profit Maximisat ion

Since we are focusing on the effects of social distinction on market behaviour, we confine

our analysis to the case of partial market coverage, i.e., hhða; p; qÞ 2 ð�hh � 1; �hh�: Monopoly

profits are given by

PM ¼ ðp � q2Þ �hhq� p þ a

qþ a

� �: ð5Þ

This expression has to be maximised with respect to the two choice variables: price and

quality.5 The first order conditions (FOCs) with respect to price and quality are

@PM

@p¼ a þ �hhq� 2p þ q2

qþ a¼ 0; ð6Þ

@PM

@q¼ �2�hhq3 þ q2ðp � a � 3a�hhÞ þ 2aqðp � aÞ þ p2 þ a�hhp � ap

ðqþ aÞ2¼ 0: ð7Þ

Figure 1. Market demand

5 The choice of maximising profits w.r.t. price or quantity is obviously irrelevant, with the same resultsobtaining in the two cases.

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Solving the system (6–7) yields

pM ¼ 3a þ 4a2 � 2a�hh þ �hh2 þ ð�hh � aÞ k

9; ð8Þ

qM ¼�hh � 4a þ k

6; ð9Þ

where k ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�hh

2 þ 16a�hh þ 16a2 � 12aq

: Observe that, positional concern being absent

(a ¼ 0), qM ¼ �hh=3 and pM ¼ 2�hh2=9 (see Lambertini, 1997). Market demand in equilibrium

amounts to

xM ¼�hh

2 � 8a�hh þ 12a � 8a2 þ ð2a þ �hhÞk3ð�hh þ 2a þ kÞ

; ð10Þ

with

xM ¼ 1 at �hh ¼ �hh0 2 þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi4a þ 1

pð11Þ

Consumer surplus is

CSM ¼Z �hh

hh½hqM þ að1 � xM Þ � pM �dh: ð12Þ

Social welfare corresponds to SW M ¼ CSM þ PM : The comparative statics properties of the

positional externality on price, quality and output are summarised by

Lemma 1: Monopoly price is always increasing in a. Moreover, xM < 1 for all �hh 2 ½1; �hh0Þ.

• lima!0 xM ¼ �hh=3; lima!1 xM ¼ 1=2:• For all �hh 2 1; 3=2½ Þ, @xM=@a > 0 and @qM=@a < 0:• For all �hh 2 ð3=2; �hh

0 Þ, @xM=@a < 0 and @qM=@a > 0:• For �hh ¼ 3=2 , @xM=@a ¼ @qM=@a ¼ 0, with xM ¼ qM ¼ 1=2.

For the sake of brevity, we omit the complete proof of Lemma 1. To verify the behaviour of

xM as a changes, it suffices to examine

@xM

@a¼ 4½30a2 � 18a � 16a3 þ 30a�hh � 24a2 �hh þ 3�hh

2 � 12a�hh2 � 2�hh

3

þ kð4a2 � 6a þ 3�hh þ 4a�hh � 2�hh2Þ�=½3kð2a þ �hh þ kÞ�2; ð13Þ

whose sign changes along �hh ¼ 3=2. The behaviour of market demand in the monopoly

equilibrium over the parameter space f�hh; ag is drawn in Figure 2.

The behaviour of market demand xM as a varies, for given levels of �hh; is represented in

Figure 3, which obtains by taking a section of the surface in Figure 2 along three specific

values of �hh 2 ½1; �hh0Þ.

Notice that, depending on the level of �hh; equilibrium quality levels tend to diverge

as the amount of positional externality a increases. This phenomenon is described in

Figure 4.

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Overall, the monopolist’s behaviour can be interpreted in the following terms:

• �hh ¼ 3=2. Here, qM ¼ xM ¼ 1=2, and the net surplus of the marginal consumer at hhis U hh ¼ hhqM � pM þ að1 � xM Þ ¼ ð1 � 2pM þ aÞ=2 ¼ 0 at pM ¼ ð1 þ aÞ=2: This

entails that, since pM increases in a as fast as the positional effect að1 � xM Þ; along�hh ¼ 3=2 the identity of the marginal consumer, and therefore the size of demand, is

independent of a. As a result, consumer surplus is constant.

• �hh 2 ð3=2; �hh0 Þ. In this range demand is larger, but, as a increases, the monopolist sup-

plies a better quality serving fewer consumers. When the marginal willingness to pay

Figure 2. Monopoly output in f�hh; ag

Figure 3. Monopoly output

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for quality is high, i.e. the market is rich, the presence of a positional effect leads to an

elitarian equilibrium, with few buyers of an expensive good characterised by a high

quality. Here, numerical calculations show that @CSM=@a < 0 for all a > 0:• �hh 2 1; 3=2½ Þ. In this range demand is low, but, as a increases, the monopolist lowers the

quality level, serving more consumers. When the marginal willingness to pay for

quality is low, i.e. the market is relatively poor, the presence of a positional effect is

simply exploited by the monopolist so as to increase price in spite of the reduction in

quality and the expansion in demand. Here, numerical calculations show that

@CSM=@a > 0 for all a > 0:

Hence, consumer surplus follows the behaviour of monopoly output. When consumers

attach a relatively low value to intrinsic quality �hh < 3=2� �

, the output increases in aoutweighing the decrease in quality and consumers are better off; when instead consumers’

valuation of quality is relatively higher �hh > 3=2� �

, the output decreases in a. In this case, the

increase in quality is not sufficient to compensate for the output contraction, and globally

consumer surplus is lower (although of course any single consumer who buys enjoys a large

positional externality).

As to profits and social welfare, the following holds:

Proposition 2: Monopoly profits are always increasing in the extent of the posi-

tional externality. Social welfare is increasing (decreasing) in a for sufficiently low (high)

levels of �hh:

Proof: The solution to @PM=@a ¼ 0 is �hh ¼ �hh0; which is the upper bound of the range for �hh

wherein partial market coverage obtains. For all �hh 2 ½1; �hh 0 Þ; @PM=@a > 0: As to the overall

Figure 4. Monopoly quality

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effect of the positional externality on social welfare, we observe that @SW M=@a is initially

positive and then negative as �hh increases (see Figure 5).6

Numerical calculations show that @SW M=@a ¼ 0 along

�hh ¼ ehh 3:8023 0:343 þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi0:0616 þ 0:263a

p� �;

with 3=2 < ehh < �hh0. Hence,

• @SW M

@a > 0 for all �hh 2 ð1; ehhÞ;• @SW M

@a < 0 for all �hh 2 ðehh; �hh 0 Þ.

Observe that the non monotonicity result we have derived for social welfare broadly

follows the behaviour of the monopoly output w.r.t. a (see Figure 3). Moreover, it reveals that

(i) when the market is relatively poor, i.e., for all �hh 2 1; 3=2½ Þ, welfare is increasing in the

level of externality because both profits and consumer surplus increase in a; (ii) for all�hh 2 ð3=2; ehhÞ; the internalisation of the positional effect by the monopolist more than balances

the decrease in consumer surplus; while (iii) the opposite holds when the market is relatively

rich, i.e., for all �hh 2 ðehh; �hh0Þ. In this range, what drives the result is the reduction in the number

of consumers who buy, combined with a reduction in individual net surplus, even though each

buyer enjoys an increasing gross surplus.7

IV. Welfare Maximisat ion

Consider now the behaviour of a benevolent social planner maximising social welfare with

respect to price and quality. We prove the following:

Figure 5. The derivative d ¼ @SW M=@a in the space fa; �hhg

6 In Figure 4, we confine the plot to the positive range of d: This entails that the flat region of the plotdenotes all the pairs a; �hh

� for which d � 0:

7 This is due to the fact that equilibrium price is convex in a; while the hedonic surplus hq is linear in aand the positional effect is concave in a. Hence the price grows at a faster pace than gross surplus as aincreases.

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Remark 1. At the social optimum, price is above marginal cost and the firm makes positive

profits for all a > 0:

Proof: The first order condition for welfare maximisation w.r.t. price is

@SW SP

@p¼ @P

@pþ @CS

@p¼ a2 þ a�hhq� 2ap � pqþ aq2 þ q3

ðqþ aÞ2¼ 0; ð14Þ

where superscript SP stands for social planning. The derivative of profits is (6) while the

derivative of consumer surplus is

@CS@p

¼ �q �hhqþ a � p� �

a þ qð Þ2¼ � qx

a þ q< 0 ð15Þ

This entails that, for any given quality, the socially optimal price is strictly lower than

monopoly price, because at the social optimum @P=@p > 0. Moreover, for all a > 0; this

also entails that the socially optimal price is higher than marginal cost. That is, as long as

partial coverage obtains, taking into account the surplus of consumers who are

positionally concerned leads the planner to keep output below the level associated with

marginal cost pricing. This involves that the firm makes positive profits.

Solving (14) yields pSP ¼ ða2 þ a�hhqþ aq2 þ q3Þ=ð2a þ qÞ: Then, solving @SW SP=@q ¼ 0

yields optimal quality

qSP ¼�hh � 8a þ h

6; ð16Þ

where h ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�hh

2 þ 32a�hh � 12a þ 64a2

q: Using qSP we obtain the socially optimal price

pSP ¼216a2 þ að16a � 10hþ 26�hhÞ þ �hh þ h

� �2h i

ð�hh � 8a þ hÞ36ð�hh þ 4a þ hÞ

: ð17Þ

By numerical methods, it can be verified that @pSP=@a > 0; and @qSP=@a > 0 always. The

equilibrium output is

xSP ¼2 �hh

2 � 16a�hh þ 12a � 32a2 þ ð4a þ �hhÞhh i

3ð4a þ �hh þ hÞ; ð18Þ

with full market coverage obtaining at �hh ¼ �hh00 2 þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi16a þ 1

p� �=2; with 3=2 < �hh

00< ehh < �hh

0;

and @xSP=@a < 0 always.

This proves the following lemmata:

Lemma 3: Under social planning, there emerges an elitarian equilibrium as a grows larger,

independently of �hh:

Unlike what happens in the profit-seeking monopoly, quality is always monotonically

increasing in a; and the planner’s behaviour is similar to the behaviour adopted by the

monopolist only when the market is rich �hh >3=2� �

.

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Lemma 4: The socially optimal policy as to the extent of market coverage establishes that

i) xSP > xM for all a; �hh�

such that xM < 1:ii) For all �hh � �hh

00, the planner serves all consumers for all a � 0:

iii) For all �hh 2 ½1; �hh 00 Þ, the planner prices some consumers out of the market for all a � 0:

As to the overall behaviour of social welfare as a varies, there emerges that @SW SP=@a ¼ 0

along the locus �hh ¼ �hh00; with @SW SP=@a > 0 for all �hh < �hh

00: As a result, we can state

Remark 2. Under social planning, social welfare is everywhere increasing in the amount

of the positional externality, as long as the market is partially covered.

V. Monopoly vs Social Planning

The behaviour of equilibrium outputs and qualities, outlined in Lemma 2, together with

Proposition 1 and Remark 1, prompts for a comparison of quality and welfare levels under the

alternative regimes of monopoly and social planning as the extent of positional externality

changes.

Consider first equilibrium output levels. Lemmata 1–3 produce the following corollary:

Corollary 5. Given �hh0> �hh

00for all a; the following holds:

• for all �hh 2 ½1; �hh 00 Þ; partial market coverage obtains under both monopoly and social

planning;

• for all �hh 2 ½�hh 00; �hh

0Þ, the monopolist excludes some consumers from purchase, while full

market coverage obtains under social planning;

• for all �hh 2 ½1; �hh 0 Þ; the monopolist undersupplies output;

• for all �hh > �hh0; all consumers are served under both regimes.

As far as qualities are concerned, a straightforward comparison of qM and qSP yields the

following:

Proposition 6: For all a; �hh�

; the monopolist supplies a lower quality than the social planner.

When a ¼ 0; Spence’s (1975) results hold: (i) under partial coverage the monopolist offers

the socially optimal quality while undersupplying output; (ii) under full coverage, the

monopolist distorts quality. Proposition 2 reveals that whenever there exists a positional

externality, quality is undersupplied at the monopoly equilibrium even when the market is

only partially covered.8 Indeed, we are going to show that the comparison between quality

levels may be misleading when there exist positional concerns. To draw policy recommen-

dations, we must evaluate the effect of marginal quality changes on social welfare in the

neighbourhood of the monopoly equilibrium.

Under the monopoly pricing rule pM ¼ q �hh þ q� �

þ a� �

=2; which obtains from (6), social

welfare is

8 Conversely, a situation where the monopolist offers a higher quality than the planner is the case ofnetwork externality, which can be seen as the opposite of a positional externality (see Lambertini andOrsini, 2001).

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SW M ¼3qþ 2að Þ q �hh � q

� �þ a

� �2

8 qþ að Þ2: ð19Þ

Now derive (19) w.r.t. q; to obtain

@SW M

@q¼

a þ �hhq� q2� �

�hh 4a2 þ 9aqþ 3q2ð Þ � 9q3 � a a þ 3qþ 8aqþ 19q2ð Þ� �

8 qþ að Þ3ð20Þ

Then, plug optimal monopoly quality (9) into (20), which can be used to verify that

@SW M

@q

����qM

¼ 0 at �hh1 ¼ 2 1 � 2að Þ3

and ð21Þ

�hh2 ¼ 2 2 3 þ 14a þ 4a2 þ 2a3ð Þ þ z 6 þ 13a � 2a2ð Þ½ �3 2 1 þ 5a þ a2ð Þ þ z 2 þ 5að Þ½ �

where

z ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4a2 þ 5a þ 1

pð22Þ

and

�hh1 < 1 for all a � 0 ð23Þ

�hh2 2 7

4; 2

� �for all a 2 ½0;1Þ

The solution �hh1 is clearly unacceptable, while the acceptable solution �hh2 is strictly lower than�hh0

where full market coverage obtains at the margin. Therefore, we have

@SW M

@q< 0 for all �hh 2 1; �hh2

� �ð24Þ

@SW M

@q> 0 for all �hh 2 ð�hh2; �hh

0�: ð25Þ

These results imply the following. In the standard model where a ¼ 0; finding that the

monopolist undersupplies quality is equivalent to saying that @SW M=@q > 0 in the

neighbourhood of qM : In the present setting, where positional externalities operate, the two

facts are not equivalent. In particular, the downward distortion of quality goes along with a

welfare loss if and only if �hh 2 ð�hh2; �hh0�. That is, if there exists an externality involving output,

the distortion of quality is not necessarily synonymous of a distortion in social welfare.

Accordingly, there emerges that a minimum (maximum) quality standard would be welfare-

improving when the market is rich (poor). This happens for the following reasons. Any

binding quality standard would decrease profits and therefore consumer surplus must increase

sufficiently to compensate for the negative effect on profits. Output, evaluated at the

monopoly pricing rule, is:

xjpM ¼q �hh � q� �

þ a

2 qþ að Þ ð26Þ

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which is decreasing in quality, in the neighbourhood of qM ; independently of �hh: This, in turn,

entails that

@ a 1 � xð Þ½ �@q

> 0 ð27Þ

i.e., the positional effect is increasing in the quality level. Hence, when �hh 2 ð�hh2; �hh0�;

consumers benefit from (i) an increase in gross surplus hq; and (ii) an increase in the

positional effect a 1 � xð Þ as quality increases. Besides their high evaluation of quality,

consumers also attach a positive value to demand contraction.

When instead �hh 2 1; �hh2

� �; i.e., the marginal evaluation of quality is low, consumers are

willing to accept a decrease in quality since it goes along with a more than proportional

decrease in price.

The foregoing discussion hinges upon the effect of quality changes on social welfare and

its components for a given positive value of a. An alternative view consists in comparing

the optima associated with the two regimes in terms of the size of the positional

externality as measured by parameter a. In order to evaluate welfare distortion, define

DSW ¼ SW SP � SW M � 0: This comparison is meaningful in the region �hh 2 ½1; �hh00Þ; where

partial market coverage obtains under both regimes. Numerical calculations show that, in this

parameter subset, @SW M=@a > @SW SP=@a. Therefore, DSW is decreasing in a. The ultimate

implication of this result is:

Proposition 7: As long as partial market coverage prevails under both monopoly and social

planning, the need for regulating the monopolist’s behaviour tends to disappear as the

positional externality becomes more relevant.

The reason for this result is different depending upon whether �hh is larger or lower than 3=2:If consumers’ interest in quality is high �hh > 3=2

� �; monopoly quality increases while output

decreases as a becomes larger. The first effect is positive, and dominates the second. The

opposite happens when �hh < 3=2: Therefore, the conventional wisdom concerning the need for

regulating a vertically differentiated monopolist does not carry over to the present setting

where positional externalities play a relevant role.

VI . Concluding Remarks

We have modelled the role of positional effects in a market for vertically differentiated

goods. We have considered a single-product monopoly with partial market coverage,

evaluating its welfare performance against the behaviour of a social planner.

The monopolist’s quality and quantity policy depends upon consumers’ hedonic evaluation

of quality. When the taste parameter is high enough, equilibrium quality increases while

quantity decreases in the level of positional effects. The opposite holds when the hedonic

parameter is relatively low. Accordingly, an elitarian equilibrium emerges only if the market is

rich enough. On the contrary, under social planning the equilibrium is always characterised by

such elitarian feature, independently of market affluency.

The comparative evaluation of the supply of quality at equilibrium reveals that the presence

of a positional concern induces the monopolist to undersupply product quality. If the

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� Blackwell Publishing Ltd/University of Adelaide and Flinders University of South Australia 2002.

positional effect disappears, our model coincide with Spence’s (1975), where monopoly

quality coincides with the socially optimal one under partial coverage.

As far as the welfare performance of the two regimes is concerned, we have shown that,

provided that partial market coverage obtains under both regimes, the welfare loss due to

monopoly power is decreasing in the extent of the positional externality, so that the scope for

regulation shrinks as the positional effect becomes more relevant.

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